The positive solution to the homogeneous space problem yields that
l2 is the only infinite-dimensional Banach space, up to
isomorphism, which is isomorphic to all its infinite-dimensional
subspaces. For a Banach space X which is not isomorphic to
l2, we investigate the problem of finding the number of
non-isomorphic infinite-dimensional subspaces of X. As a
consequence of our construction we also obtain a structural result
about Banach spaces containing an unconditional basic sequence.

We say a multifunction T : X® 2X* is monotone
provided that for all x,yÎX, and for all x*ÎT(x), y*ÎT(y),

áy-x,y*-x*ñ³ 0,

and we say that T is maximal monotone if its graph is not
properly included in any other monotone graph. The convex
subdifferential in Banach space and a skew linear matrix are
the canonical examples of maximal monotone multifunctions. Maximal
monotone operators play an important role in functional analysis,
optimization and partial differential equation theory, with
applications in subjects such as mathematical economics and robust
control.

In this talk, largely based on [1], I shall show how-thanks largely
to a long-neglected observation of Fitzpatrick-the originally quite
complex theory of monotone operators can be almost entirely reduced to
convex analysis. I shall also highlight various long-standing open
questions to which these new techniques have offered new access, [2],
[3].

The Morse-Sard theorem states that if f : Rn®Rm is a Cr smooth function, with r > max{n-m,0}, and Cf
stands for the set of critical points of f (that is, the points of
X at which the differential of f is not surjective), then the set
of critical values, f(Cf), is of (Lebesgue) measure zero in Rm.
This result is no longer true for functions defined on infinite
dimensional Banach spaces. However, under this setting it is possible
to develop a strong approximate version of such classical
principle.

I was totally surprised that all of my earliest research in the 1970s
has been used intensively in the recent development of quantum
computers. By all means, I need to seek the new meanings of the old
values of completely positive linear maps in the setup of
non-commutative matrices.

The relative position of two 1-dimensional subspaces of Hilbert space
is expressed fully by a single number in [0 1]. But suppose we
study subspaces P spanned by orthonormal p1, p2, ..., pk and
Q spanned by orthonormal q1, q2, ..., qk for higher k. To
express the relative position of P and Q fully requires an
unordered k-tuple of numbers; this theory has been understood since
the 1880s and generalized to infinite-dimensional subspaces. In
contrast, it is known that the relative position of three
k-dimensional subspaces can not be expressed fully by a manageable
invariant.

Nevertheless the manifold of k-subspaces can be studied to
advantage. Here is a typical example of a question which can be asked
of three subspaces and may have illuminating answers: Given subspaces
P, Q, R, is P closer to Q than R is? This talk gives a modern
way of dealing with such questions.

Let L(a) be an operator function of the class C1 ([a,b], S(H) ) such that L(a) << 0, L(b) >> 0 and for all
xÎH \{0} the function ( L(a)x, x )
has exactly one zero p(x) in (a,b). Define the following
nonlinear operator

Tx =

ìïí
ïî

L

æè

p(x)

öø

x,

x¹ 0,

0,

x = 0.

We study a connection between solvability problems for the equation
Tx=y and completeness problems for eigenvectors of the operator
functions L(a). We give some sufficient conditions for
completeness of eigenvectors corresponding to eigenvalues from the
interval [a, b] which are based on separation properties of the
functional p(x).

In 2002 Gluskin and I showed that a polytope with few vertices is far
from being symmetric in a sense of Banach-Mazur distance. More
precisely, it was shown that Banach-Mazur distance between such a
polytope and any symmetric convex body is large. In this talk we
introduce a new, averaging-type parameter to measure the asymmetry.
It turns out that, surprisingly, this new parameter is still very
large, in fact it satisfies the same lower bound as Banach-Mazur
distance. We apply our results to provide a lower estimate on the
vertex index of a symmetric convex body, which was recently introduced
in by Bezdek and myself. Furthermore, we give the affirmative answer
to a conjecture by K. Bezdek on the quantitative illumination
problem.

We investigate the properties and structure of topologically
transitive multiplicative semigroups of real or complex matrices, and
are particularly interested in the question: "What extra conditions
must be imposed on such semigroups to guarantee transitivity?"

A set S of matrices is topologically transitive if any
non-zero vector can be mapped arbitrarily close to any other vector by
a matrix in S, and is transitive if any non-zero vector
can be mapped extactly to any other vector by a matrix in
S.

This talk is based on joint work with Leo Livshits and Heydar Radjavi.

I will discuss some results on irreducible semigroups of matrices that
Heydar Radjavi and I have recently obtained, and consider which of
these results might possibly generalize to semigroups of operators on
Hilbert space.

We have established the following: a transitive, closed, homogeneous
semigroup of linear transformations on a finite-dimensional space
either has zero divisors or is simultaneously similar to a group
consisting of scalar multiples of unitary transformations. The proof
begins by establishing the result that for each closed homogeneous
semi-group with no zero divisors there is a k such that the spectral
radius of AB is at most kr(A) r(B) for all A and B in the
semigroup. (It is also shown that the spectral radius is not
k-submultiplicative on any transitive semigroup of compact
operators.)

We have also proven that an irreducible semigroup of complex matrices
is, respectively, finite, countable or bounded if there is a non-zero
linear functional whose range on the semigroup has the corresponding
property. Moreover, an irreducible semigroup is finite, countable or
bounded if it has a nonzero ideal with the corresponding property.

We present constructive methods to obtain compactly supported
biorthogonal wavelets adapted to differential (pseudodifferential)
operators. Such wavelets provide a diagonal form of the corresponding
operators and can be useful for numerical applications. Note that the
construction of the wavelets boils down mainly to obtain the
corresponding masks. Note also that operator-adapted wavelets are
closely connected with wavelet bases for some functional spaces (in
particular, Sobolev's spaces). We introduce an original approach to
obtain wavelets on an interval adapted to monomial differential
operators. A generalization to sum of differential operators is
discussed. We present a method to adapt wavelets to differential
operators with polynomial coefficients. In the multidimensional case
we introduce a construction of wavelets adapted to hyperbolic
differential operators. A preliminary idea on how to adapt wavelets
to the Laplace operator is also considered. Note that, in general,
operator-adapted wavelets are not shift and scale invariant (second
generation wavelets) and this raises the problem of the well-posedness
of the corresponding Multiresolution Analyses. We also discuss a
conceptual possibility of adapting frames to operators.